SOLUTION MANUAL
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets and Relations 1
I. Groups and Subgroups
1. Introduction and Examples 4
2. Binary Operations 7
3. Isomorphic Binary Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
7. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
8. Groups of Permutations 26
9. Orbits, Cycles, and the Alternating Groups 30
10. Cosets and the Theorem of Lagrange 34
11. Direct Products and Finitely Generated Abelian Groups 37
12. Plane Isometries 42
III. Homomorphisms and Factor Groups
13. Homomorphisms 44
14. Factor Groups 49
15. Factor-Group Computations and Simple Groups 53
16. Group Action on a Set 58
17. Applications of G-Sets to Counting 61
IV. Rings and Fields
18. Rings and Fields 63
19. Integral Domains 68
20. Fermat’s o and o Euler’s o Theorems 72
21. The o Field o of o Quotients o of o an o Integral o Domain 74
22. Rings o of o Polynomials 76
23. Factorization oof oPolynomials oover oa oField 79
24. Noncommutative o Examples 85
25. Ordered o Rings o and o Fields 87
V. Ideals o and o Factor o Rings
26. Homomorphisms o and o Factor oRings 89
27. Prime oand oMaximal oIdeals 94
28. Gröbner oBases ofor oIdeals 99
, VI. Extension o Fields
29. Introduction o to o Extension o Fields 103
30. Vector o Spaces 107
31. Algebraic o Extensions 111
32. Geometric oConstructions 115
33. Finite o Fields 116
VII. Advanced o Group o Theory
34. Isomorphism oTheorems 117
35. Series oof oGroups 119
36. Sylow o Theorems 122
37. Applications o of o the o Sylow o Theory 124
38. Free o Abelian o Groups 128
39. Free oGroups 130
40. Group o Presentations 133
VIII. Groups o in o Topology
41. Simplicial o Complexes o and o Homology o Groups 136
42. Computations o of o Homology o Groups 138
43. More o Homology o Computations o and o Applications 140
44. Homological o Algebra 144
IX. Factorization
45. Unique o Factorization o Domains 148
46. Euclidean o Domains 151
47. Gaussian o Integers o and o Multiplicative o Norms 154
X. Automorphisms o and o Galois o Theory
48. Automorphisms o of o Fields 159
49. The o Isomorphism o Extension o Theorem 164
50. Splitting o Fields 165
51. Separable oExtensions 167
52. Totally oInseparable oExtensions 171
53. Galois o Theory 173
54. Illustrations oof oGalois oTheory 176
55. Cyclotomic oExtensions 183
56. Insolvability o of o the o Quintic 185
APPENDIX o Matrix o Algebra 187
iv
, 0. o Sets oand oRelations 1
0. Sets o and o Relations
√ √
1. { 3, o − 3} 2. o The o set o is o empty.
3. o{1, o−1, o2, o−2, o3, o−3, o4, o−4, o5, o−5, o6, o−6, o10, o−10, o12, o−12, o15, o−15, o20, o−20, o30, o−30,
60, o−60}
4. o {−10, o−9, o−8, o−7, o−6, o−5, o−4, o−3, o−2, o−1, o0, o1, o2, o3, o4, o5, o6, o7, o8, o9, o10, o11}
5. It ois onot oa owell-defined oset. o (Some omay oargue othat ono oelement oof oZ+ ois olarge, obecause
oevery oelement oexceeds oonly oa ofinite onumber oof oother oelements obut ois oexceeded oby oan oinfinite
onumber oof oother oelements. oSuch opeople omight oclaim othe oanswer oshould obe o∅.)
6. ∅ 7. o The o set o is o ∅ o because o 33 o= o27 o and o 43 o= o64.
8. o It o is o not o a o well-defined o set. 9. o Q
10. The o set o containing o all o numbers o that o are o (positive, o negative, o or o zero) o integer o multiples
o of o 1, o 1/2, o or o1/3.
11. {(a, o1), o(a, o2), o(a, oc), o (b, o1), o(b, o2), o(b, oc), o(c, o1), o(c, o2), o(c, oc)}
12. a. o It ois oa ofunction. o It ois onot oone-to-one osince othere oare otwo opairs owith osecond omember o4.
o It o is o not o onto
B o because othere ois ono opair owith osecond omember o2.
b. (Same o answer o as o Part(a).)
c. It o is o not o a o function o because o there o are o two o pairs o with o first o member o 1.
d. It o is o a o function. o It o is o one-to-one. o It o is o onto o B o because o every o element o of o B
o appears o as o second omember oof osome opair.
e. It ois oa ofunction. o It ois onot oone-to-one obecause othere oare otwo opairs owith osecond
omember o6. o It ois onot oonto oB obecause othere ois ono opair owith osecond omember o2.
f. It ois o not o a o function o because o there o are o two o pairs o with o first o member o 2.
13. Draw o the o line o through o P o and o x, o and o let o y o be o its o point o of o intersection o with o the o line
o segment o CD.
14. a. o φ o: o[0, o1] o→ o [0, o2] o where o φ(x) o= o2x b. o φ o: o[1, o3] o → o [5, o25] o where o φ(x) o= o5 o+ o10(x o−
o1)
c. o φ o: o[a, →
ob] [c, od] o where o φ(x) o = o c o+ o−d−c o(x a)
b−a
1o
15. Let o φ o: oS o → oR o be o defined o by o φ(x) o= otan(π(x
2
o− o )).
16. a. o ∅; o cardinality o 1 b. o ∅, o{a}; o cardinality o 2 c. o ∅, o{a}, o{b}, o{a, ob}; o cardinality o 4
d. o ∅, o{a}, o{b}, o{c}, o{a, ob}, o{a, oc}, o{b, oc}, o{a, ob, oc}; o cardinality o 8
17. Conjecture: |P(A)| o= o2s o = o2|A|.
Proof oThe onumber oof osubsets oof oa oset oA odepends oonly oon othe ocardinality oof oA, onot oon
owhat othe oelements oof o A o actually o are. o Suppose oB o= o{1, o2, o3, o· o· o· o, os o− o1} o and o A o=
o{1, o2, o3, o o , os}. o Then o A o has o all
the oelements oof oB oplus othe oone oadditional oelement os. o All osubsets oof oB oare oalso
osubsets oof oA; othese oare oprecisely othe osubsets oof oA othat odo onot ocontain os, oso othe
onumber oof osubsets oof oA onot ocontaining os ois o|P(B)|. o Any oother osubset oof oA omust
First Course in Abstract Algebra A
8th Edition by John B. Fraleigh
All Chapters Full Complete
, CONTENTS
0. Sets and Relations 1
I. Groups and Subgroups
1. Introduction and Examples 4
2. Binary Operations 7
3. Isomorphic Binary Structures 9
4. Groups 13
5. Subgroups 17
6. Cyclic Groups 21
7. Generators and Cayley Digraphs 24
II. Permutations, Cosets, and Direct Products
8. Groups of Permutations 26
9. Orbits, Cycles, and the Alternating Groups 30
10. Cosets and the Theorem of Lagrange 34
11. Direct Products and Finitely Generated Abelian Groups 37
12. Plane Isometries 42
III. Homomorphisms and Factor Groups
13. Homomorphisms 44
14. Factor Groups 49
15. Factor-Group Computations and Simple Groups 53
16. Group Action on a Set 58
17. Applications of G-Sets to Counting 61
IV. Rings and Fields
18. Rings and Fields 63
19. Integral Domains 68
20. Fermat’s o and o Euler’s o Theorems 72
21. The o Field o of o Quotients o of o an o Integral o Domain 74
22. Rings o of o Polynomials 76
23. Factorization oof oPolynomials oover oa oField 79
24. Noncommutative o Examples 85
25. Ordered o Rings o and o Fields 87
V. Ideals o and o Factor o Rings
26. Homomorphisms o and o Factor oRings 89
27. Prime oand oMaximal oIdeals 94
28. Gröbner oBases ofor oIdeals 99
, VI. Extension o Fields
29. Introduction o to o Extension o Fields 103
30. Vector o Spaces 107
31. Algebraic o Extensions 111
32. Geometric oConstructions 115
33. Finite o Fields 116
VII. Advanced o Group o Theory
34. Isomorphism oTheorems 117
35. Series oof oGroups 119
36. Sylow o Theorems 122
37. Applications o of o the o Sylow o Theory 124
38. Free o Abelian o Groups 128
39. Free oGroups 130
40. Group o Presentations 133
VIII. Groups o in o Topology
41. Simplicial o Complexes o and o Homology o Groups 136
42. Computations o of o Homology o Groups 138
43. More o Homology o Computations o and o Applications 140
44. Homological o Algebra 144
IX. Factorization
45. Unique o Factorization o Domains 148
46. Euclidean o Domains 151
47. Gaussian o Integers o and o Multiplicative o Norms 154
X. Automorphisms o and o Galois o Theory
48. Automorphisms o of o Fields 159
49. The o Isomorphism o Extension o Theorem 164
50. Splitting o Fields 165
51. Separable oExtensions 167
52. Totally oInseparable oExtensions 171
53. Galois o Theory 173
54. Illustrations oof oGalois oTheory 176
55. Cyclotomic oExtensions 183
56. Insolvability o of o the o Quintic 185
APPENDIX o Matrix o Algebra 187
iv
, 0. o Sets oand oRelations 1
0. Sets o and o Relations
√ √
1. { 3, o − 3} 2. o The o set o is o empty.
3. o{1, o−1, o2, o−2, o3, o−3, o4, o−4, o5, o−5, o6, o−6, o10, o−10, o12, o−12, o15, o−15, o20, o−20, o30, o−30,
60, o−60}
4. o {−10, o−9, o−8, o−7, o−6, o−5, o−4, o−3, o−2, o−1, o0, o1, o2, o3, o4, o5, o6, o7, o8, o9, o10, o11}
5. It ois onot oa owell-defined oset. o (Some omay oargue othat ono oelement oof oZ+ ois olarge, obecause
oevery oelement oexceeds oonly oa ofinite onumber oof oother oelements obut ois oexceeded oby oan oinfinite
onumber oof oother oelements. oSuch opeople omight oclaim othe oanswer oshould obe o∅.)
6. ∅ 7. o The o set o is o ∅ o because o 33 o= o27 o and o 43 o= o64.
8. o It o is o not o a o well-defined o set. 9. o Q
10. The o set o containing o all o numbers o that o are o (positive, o negative, o or o zero) o integer o multiples
o of o 1, o 1/2, o or o1/3.
11. {(a, o1), o(a, o2), o(a, oc), o (b, o1), o(b, o2), o(b, oc), o(c, o1), o(c, o2), o(c, oc)}
12. a. o It ois oa ofunction. o It ois onot oone-to-one osince othere oare otwo opairs owith osecond omember o4.
o It o is o not o onto
B o because othere ois ono opair owith osecond omember o2.
b. (Same o answer o as o Part(a).)
c. It o is o not o a o function o because o there o are o two o pairs o with o first o member o 1.
d. It o is o a o function. o It o is o one-to-one. o It o is o onto o B o because o every o element o of o B
o appears o as o second omember oof osome opair.
e. It ois oa ofunction. o It ois onot oone-to-one obecause othere oare otwo opairs owith osecond
omember o6. o It ois onot oonto oB obecause othere ois ono opair owith osecond omember o2.
f. It ois o not o a o function o because o there o are o two o pairs o with o first o member o 2.
13. Draw o the o line o through o P o and o x, o and o let o y o be o its o point o of o intersection o with o the o line
o segment o CD.
14. a. o φ o: o[0, o1] o→ o [0, o2] o where o φ(x) o= o2x b. o φ o: o[1, o3] o → o [5, o25] o where o φ(x) o= o5 o+ o10(x o−
o1)
c. o φ o: o[a, →
ob] [c, od] o where o φ(x) o = o c o+ o−d−c o(x a)
b−a
1o
15. Let o φ o: oS o → oR o be o defined o by o φ(x) o= otan(π(x
2
o− o )).
16. a. o ∅; o cardinality o 1 b. o ∅, o{a}; o cardinality o 2 c. o ∅, o{a}, o{b}, o{a, ob}; o cardinality o 4
d. o ∅, o{a}, o{b}, o{c}, o{a, ob}, o{a, oc}, o{b, oc}, o{a, ob, oc}; o cardinality o 8
17. Conjecture: |P(A)| o= o2s o = o2|A|.
Proof oThe onumber oof osubsets oof oa oset oA odepends oonly oon othe ocardinality oof oA, onot oon
owhat othe oelements oof o A o actually o are. o Suppose oB o= o{1, o2, o3, o· o· o· o, os o− o1} o and o A o=
o{1, o2, o3, o o , os}. o Then o A o has o all
the oelements oof oB oplus othe oone oadditional oelement os. o All osubsets oof oB oare oalso
osubsets oof oA; othese oare oprecisely othe osubsets oof oA othat odo onot ocontain os, oso othe
onumber oof osubsets oof oA onot ocontaining os ois o|P(B)|. o Any oother osubset oof oA omust
